Does anybody know an easy explanation of the proof of Artin's vanishing theorem (that the etale cohomology of an affine variety of dimension $n$ over an algebraically closed field vanishes in degrees $>n$, or of any other version of this statement)? I have found some proofs; all of them are step by step, and it is not clear to me which of these steps are the most important ones. So, what is the central idea here?

I was curious myself after learning this result sometime ago from Lazarsfeld's book on positivity (he calls it the ArtinGrothendieck theorem). The corresponding statement for smooth varieties over the complex numbers and singular cohomology (theorem of AndreottiFrankel) follows from the fact that Morse theory shows that the variety is homotopy equivalent to a CWcomplex with no cells in dimensions $>n$. The etale cohomology counterpart works more generally for constructible sheaves. This is probably not very helpful, but here is a sketch of the argument from Lazarsfeld (he uses constructible sheaves in the complex topology, but it should adapt to etale sheaves):
So all in all it is a typical example of devissage, which I usually to dislike but slowly learn to appreciate. I think from the outline it is clear which are the key ideas, but I would still really like to see a conceptual proof. 

